Let $\zeta \in \mathbb{C} \text { be a } 5 \text { -th primitive root of unity, the Galois group } \operatorname{Gal}(\mathbb{Q}(\zeta) / \mathbb{Q})\cong (\mathbb{Z} / 5 \mathbb{Z})^{*}=\left\{e, \alpha, \alpha^{2}, \alpha^{3}\right\}.$
I want to find he subfield that corresponds to the unique order 2 subgroup of it, and write it in-terms of $\mathbb{Q}(\lambda),$ where $\lambda$ is square root some integers.
My approach: The unique order $2$ subgroup is $H=\{e,\alpha^2\}.$ Then, the fundamental theorem of Galois theory says, there is an inclusion-reversing bijection correspondence $H\to \mathbb{Q(\zeta)}^H$. However, how can I write $\mathbb{Q(\zeta)}^H$ in $\mathbb{Q}(\lambda)$? We didn't learn the spider net in lessons :( So is there any way to find the $\lambda$ without drawing picture?
Hint. First prove that $\mathbb{Q}(\zeta)^H=\mathbb{Q}(\zeta+\zeta^{-1})$ using definitions.
Then notice that $\zeta+\zeta^{-1}=2\cos(2\pi/5)$, then deduce that you can take $\lambda=\sqrt{5}$.